Optimal Investment in a Market with Borrowing and a Combined Interest Rate Model

We consider the problem of optimal investment in a market with borrowing, a stochastic interest rate, and the power utility from terminal wealth. A certain combined Hull-White and quadratic-affine interest rate model is introduced, which in particular renders the market incomplete in general. The resulting problem, due to the higher interest rate for borrowing than for lending, is an optimal stochastic control problem with a nonlinear system dynamics and an unbounded coefficient. The explicit closed-form solution is found as a linear state-feedback control the gain of which can have up to three different regimes.